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Unformatted text preview: Homework 6 Solutions Math 471, Fall 2006 Assigned: Friday, October 20, 2006 Due: Friday, October 27, 2006 • Include a cover page • Clearly label all plots using title , xlabel , ylabel , legend • Use the subplot command to compare multiple plots • Include printouts of all Matlab code, labeled with your name, date, section, etc. (1) (Condition Numbers) P. 187 #1, 2 Part (a): κ ( AB ) =  AB  ·  B 1 A 1  ≤  A  ·  A 1  ·  B  ·  B 1  = κ ( A ) κ ( B ). Part (b): κ ( αA ) =  αA  ·  1 α A 1  =  α   α   A  ·  A 1  = κ ( A ). (2) (Error Estimates) P. 188 #7c, 8c 1 1 1 1 1 1 x = 2 , x = 2 2 , ˜ x = 1 . 9000 2 . 1000 . 1000  x  ∞ = 2 , e = x ˜ x = . 1000 . 1000 . 1000 ,  e  ∞ = 0 . 1 r = b A ˜ x = . 1000 . 2000 . 1000 ,  r  ∞ = 0 . 2 ,  b  ∞ = 2  A  ∞ = 3 ,  A 1  ∞ = 4 , κ ( A ) = 12  e  ∞  x  ∞ = . 1 2 = κ ( A ) 24  r  ∞  b  ∞ ≤ κ ( A )  r  ∞  b  ∞ = 12 . 2 2 = 12 · (0 . 1) (3) (Positive Definite and Strictly Diagonally Dominant Matrices) P. 220 #3 Part (a) : A matrix is positive definite if and only if the determinants of the principal submatrices are all positive. Δ 1 = a > , Δ 2 = 4 a 1 > ⇒ a > 1 4 Δ 3 = det( A ) = 19 a 5 > ⇒ a > 5 19 The intersection of these three intervals is a > 5 / 19 which extends the interval a > 1 we obtained using the definition of the positive definite matrix. Therefore, ifpositive definite matrix....
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This homework help was uploaded on 04/02/2008 for the course MATH 471 taught by Professor Linzhi during the Winter '08 term at University of Michigan.
 Winter '08
 LinZhi
 Math, Numerical Analysis, Cholesky Decomposition

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